Point-Slope Formula: y − y₁ = m(x − x₁), Explained
Learn what the point-slope formula means, how to use it in three quick steps, and try the calculator that shows every step. It’s the fastest way to write a line when you know a point and the slope.
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The point-slope formula is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line. You use it to write the equation of a straight line the moment you know one point and the slope, without solving for the y-intercept first.
What Is Point-Slope Form?
Point-slope form is one of the three main ways to write the equation of a straight line. The other two are slope-intercept form (y = mx + b) and standard form (Ax + By = C). What makes point-slope special is that it’s built directly from two things you very often already have: a single point on the line and the line’s slope.
Here’s the formula again, and it’s worth memorizing because you’ll use it constantly in algebra:
y − y₁ = m(x − x₁)
If you’ve got a point and a slope, you don’t need to do any rearranging to get started. You just substitute and you’re done. That’s why it’s usually the first form you’ll reach for when a problem hands you a point and a slope.
When to reach for it. The moment a problem gives you a point and a slope, or two points, point-slope form lets you write the equation in a single step, with no y-intercept needed.
The Point-Slope Formula Explained
Each piece of the formula has a specific job. Once you see what each symbol stands for, the whole thing stops looking abstract:
- m is the slope of the line, or how steep it is.
- (x₁, y₁) is a known point the line passes through. The little 1 means “the first point,” and those are real numbers you plug in.
- x and y stay as variables. Don’t substitute numbers for them, because they represent every point on the line.
Watch the signs. The formula has minus signs baked in, so a negative coordinate flips to a plus. For the point (3, −2), y − (−2) becomes y + 2, not y − 2. It’s the most common slip, and we’ll flag it again in the examples.
How to Use the Point-Slope Formula
Identify the point
Write your known point as (x1, y1). If you have two points, you can use either one.
Identify the slope
Note the slope m. If it is not given, find it from two points first.
Substitute and simplify
Drop the point and slope into y - y1 = m(x - x1). That is your equation.
Point-Slope Formula Derivation
The formula isn’t something you have to take on faith. It comes straight from the definition of slope. Slope between any point (x, y) on the line and your fixed point (x₁, y₁) is:
m = (y − y₁) / (x − x₁)
Now multiply both sides by (x − x₁) to clear the fraction, and you land right on the point-slope formula:
y − y₁ = m(x − x₁)
That’s the entire derivation. It’s really just the slope definition, rearranged so nothing sits in a denominator.
Worked Examples
Example 1: From a Point and a Slope
Suppose a line passes through (3, −2) with a slope of 5. Substitute x₁ = 3, y₁ = −2, and m = 5:
y − (−2) = 5(x − 3) → y + 2 = 5(x − 3)
Notice how subtracting −2 became “plus 2.” That’s the sign rule in action.
Example 2: From Two Points
Say the line goes through (1, 2) and (4, 8). First find the slope: m = (8 − 2) / (4 − 1) = 6 / 3 = 2. Now pick (1, 2) as your point:
y − 2 = 2(x − 1)
If you’d picked (4, 8) instead, you’d get y − 8 = 2(x − 4). They look different, but they’re the same line. Here’s the full two-point walkthrough.
Example 3: Vertical Line (Undefined Slope)
If two points share the same x-value, like (2, 1) and (2, 7), the slope is undefined and you can’t use point-slope form. The line is vertical, so you just write x = 2. Every point-slope problem quietly assumes the slope exists.
Converting Point-Slope Form
Once you’ve written point-slope form, it’s easy to convert it into the other two forms. Here’s the short version, and there’s a complete conversion guide if you want every step.
To Slope-Intercept Form (y = mx + b)
Just solve for y. Take y + 2 = 5(x − 3), distribute the 5 to get y + 2 = 5x − 15, then subtract 2: y = 5x − 17. Now you can read the y-intercept straight off.
To Standard Form (Ax + By = C)
Move the x and y terms to one side and keep the constant on the other, using integer coefficients with A positive. From y = 5x − 17 you get 5x − y = 17.
Point-Slope vs Slope-Intercept vs Standard
Each form shines in a different situation. This table shows what each one gives you at a glance, and our point-slope vs slope-intercept comparison goes deeper.
| Form | Looks like | Best when you know |
|---|---|---|
| Point-slope | y − y₁ = m(x − x₁) | A point and the slope |
| Slope-intercept | y = mx + b | The slope and the y-intercept |
| Standard | Ax + By = C | You need integer coefficients |
Real-World Applications
Point-slope form isn’t just a classroom exercise. Any time something changes at a steady rate from a known starting value, you’ve got a point and a slope. A taxi that charges a flat pickup fee plus a rate per mile, a savings balance that grows by a fixed amount each week, or a phone plan with a base cost plus a per-gigabyte charge all fit the pattern. You know one data point and the rate of change, so point-slope writes the model in a single step.